Electron and hole spectra in quantum wire with two quantum dots in the electric field
The energy spectrum of electron and hole is investigated in a complicated nanoheterosystem consisting of two cylindrical semiconductor quantum dots placed into semiconductor quantum wire. Quantum dots are separated by barrier-layer, which is under the action of a constant electric field. The depen...
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Інститут фізики конденсованих систем НАН України
2007
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | Electron and hole spectra in quantum wire with two quantum dots in the electric field / O. Makhanets, A. Gryschyk, M. Dovganiuk // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 69-74. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1180612017-05-29T03:04:32Z Electron and hole spectra in quantum wire with two quantum dots in the electric field Makhanets, O. Gryschyk, A. Dovganiuk, M. The energy spectrum of electron and hole is investigated in a complicated nanoheterosystem consisting of two cylindrical semiconductor quantum dots placed into semiconductor quantum wire. Quantum dots are separated by barrier-layer, which is under the action of a constant electric field. The dependencies of electron and hole energies on geometric parameters of quantum dots and electric field intensity are analysed. Дослiджено енергетичний спектр електрона i дiрки в складнiй наногетеросистемi, що складається з двох цилiндричних напiвпровiдникових квантових точок, розташованих у напiвпровiдниковому квантовому дротi. Квантовi точки роздiленi шаром-бар’єром, до якого прикладене постiйне електричне поле. Проаналiзовано залежностi енергiй електрона i дiрки вiд геометричних розмiрiв квантових точок i величини напруженостi електричного поля. 2007 Article Electron and hole spectra in quantum wire with two quantum dots in the electric field / O. Makhanets, A. Gryschyk, M. Dovganiuk // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 69-74. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 73.21.Fg, 73.21.Hb, 73.21.La DOI:10.5488/CMP.10.1.69 http://dspace.nbuv.gov.ua/handle/123456789/118061 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| description |
The energy spectrum of electron and hole is investigated in a complicated nanoheterosystem consisting of
two cylindrical semiconductor quantum dots placed into semiconductor quantum wire. Quantum dots are
separated by barrier-layer, which is under the action of a constant electric field. The dependencies of electron
and hole energies on geometric parameters of quantum dots and electric field intensity are analysed. |
| format |
Article |
| author |
Makhanets, O. Gryschyk, A. Dovganiuk, M. |
| spellingShingle |
Makhanets, O. Gryschyk, A. Dovganiuk, M. Electron and hole spectra in quantum wire with two quantum dots in the electric field Condensed Matter Physics |
| author_facet |
Makhanets, O. Gryschyk, A. Dovganiuk, M. |
| author_sort |
Makhanets, O. |
| title |
Electron and hole spectra in quantum wire with two quantum dots in the electric field |
| title_short |
Electron and hole spectra in quantum wire with two quantum dots in the electric field |
| title_full |
Electron and hole spectra in quantum wire with two quantum dots in the electric field |
| title_fullStr |
Electron and hole spectra in quantum wire with two quantum dots in the electric field |
| title_full_unstemmed |
Electron and hole spectra in quantum wire with two quantum dots in the electric field |
| title_sort |
electron and hole spectra in quantum wire with two quantum dots in the electric field |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118061 |
| citation_txt |
Electron and hole spectra in quantum wire with two quantum dots in the electric field / O. Makhanets, A. Gryschyk, M. Dovganiuk // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 69-74. — Бібліогр.: 12 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT makhanetso electronandholespectrainquantumwirewithtwoquantumdotsintheelectricfield AT gryschyka electronandholespectrainquantumwirewithtwoquantumdotsintheelectricfield AT dovganiukm electronandholespectrainquantumwirewithtwoquantumdotsintheelectricfield |
| first_indexed |
2025-07-08T13:17:59Z |
| last_indexed |
2025-07-08T13:17:59Z |
| _version_ |
1837084896952057856 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 1(49), pp. 69–74
Electron and hole spectra in quantum wire with two
quantum dots in the electric field
O.Makhanets, A.Gryschyk, M.Dovganiuk
Fedkovych Chernivtsi National University, 2, Kotsyubinskoho Str., Chernivtsi, 58012, Ukraine∗
Received September 4, 2006
The energy spectrum of electron and hole is investigated in a complicated nanoheterosystem consisting of
two cylindrical semiconductor quantum dots placed into semiconductor quantum wire. Quantum dots are
separated by barrier-layer, which is under the action of a constant electric field. The dependencies of electron
and hole energies on geometric parameters of quantum dots and electric field intensity are analysed.
Key words: quantum wire, quantum dot, energy spectrum
PACS: 73.21.Fg, 73.21.Hb, 73.21.La
1. Introduction
Artificial atoms or quantum dots (QDs) constructed from semiconductors are expected to pro-
vide the basis for future generations of device technologies such as threshholdless lasers and ultra-
dense memories. When several quantum dots are connected to each other, they provide remarkable
phenomena due to the interplay of electron correlations, interference effects, etc, which depend on
how the dots are arranged: e.g. double quantum dots coupled in series or parallel [1–5].
In this context double quantum dots are of particular interest from two different viewpoints
such as implementation of quantum bits (qubit) [6,7] and a model system for molecular binding
under controlled conditions [8,9].
” ”0”1” ”1”
0
h
1 h 2
” ”0 ” ”0
Z
U(z) ”2”
”2”
”2”
”2”
-h1 +h2
F
U0
Eg
HgS
HgSCdS HgSCdS
e
U0
h
CdS
Figure 1. Geometrical scheme of a
nanosystem and the dependence of
electron and hole potential energy on z
variable.
At present, of particular interest are the double
quantum dots embedded into quantum nanowires.
Such systems have already been created and are
being intensively investigated experimentally [10,11].
The theory of spectra and wave functions of quasi-
particles (electrons, holes, excitons) in quantum dots
embedded into the quantum wires is only at the start
of development [12]. Such combined nanoheterosys-
tems are rather complicated for mathematical de-
scription due to the complicated fitting conditions
for the wave functions of quasiparticles.
In this paper we are going to study the stationary
energy states of electron and hole in a nanoheterosys-
tem consisting of two semiconductor quantum dots
(of different size) embedded into the cylindrical quan-
tum wire. The thin barrier-layer separating quantum
dots are under the action of a constant electric field
of a fixed intensity (~F ), directed along the axial axis
of nanohetrosystem (figure 1).
∗E-mail: theorphys@chnu.cv.ua
c© O.Makhanets, A.Gryschyk, M.Dovganiuk 69
O.Makhanets, A.Gryschyk, M.Dovganiuk
2. The theory of electron (hole) energy spectrum in cylindri cal quantum
wire with two quantum dots inside
The complicated semiconductor cylindrical quantum wire (“0”), containing two quantum dots
of the same material (“1”), separated by a thin layer of another material (“0”) is under research.
The radius of QW (ρ0), the height of QDs (h1 and h2), respectively and thickness of the layer
separating QDs (∆) are assumed to be fixed (figure 1).
A complicated quantum wire is placed into the external medium forming the infinite potential
barriers for the electron (hole). The electron (hole) effective masses are different in different parts
of a nanosystem:
µeh(z) =
{
µeh
0 , medium “0”,
µeh
1 , medium “1”.
(1)
It is also assumed that the lattice constants a0 and a1 of the media “0” and “1” are to be
close in magnitude. For example, the computer calculations are performed for the nanosystem
created at the base of β − HgS and β − CdS crystals where the lattice constants are so close that
(a1 − a0)/a0 6 1%. Since the interfaces between different parts of nanosystem are rather straight,
it is possible to use the approximation of rectangular potential energies for the electron and hole.
Taking this into account as well as the fact that the constant electric field with intensity of ~F
is applied to the barrier-layer (“0”) like it is shown in figure 1, the potential energy can be written
in the form
U e (ρ, ϕ, z) =
U e
0 , z < −h1 and z > ∆ + h2 ,
0, −h1 6 z 6 0 and ∆ 6 z 6 ∆ + h2 ,
U e
0 − eFz, 0 6 z 6 ∆,
(2)
Uh (ρ, ϕ, z) =
−EHgS
g − Uh
0 , z < −h1 and z > ∆ + h2 ,
−EHgS
g , −h1 6 z 6 0 and ∆ 6 z 6 ∆ + h2 ,
−EHgS
g − Uh
0 + eFz, 0 6 z 6 ∆ ,
(3)
where e is the electron charge, U eh
0 = V eh
0 −V eh
1 , V e
0 , V e
1 (V h
0 , V h
1 ) are the electron (hole) potential
energies in the corresponding media “0” and “1”, taken respectively in vacuum.
The analytical calculation of electron and hole energy spectra and wave functions are equal,
since the further formulas are presented for the electron omitting index e.
In order to investigate the electron quantum states, it is necessary to solve the Schrodinger
equation.
_
HΨ(~r) = EΨ(~r) (4)
with Hamiltonian
_
H = −
~
2
2
~∇
1
µ (z)
~∇ + U (ρ, ϕ, z) . (5)
Taking into account the cylindrical symmetry it is convenient to write the wave function Ψ (r̃)
as in [12]
Ψnρm (~r) =
(
−πρ2
0Jm−1
(
χnρm
)
Jm+1
(
χnρm
))−1/2
Jm
(
χnρm
ρ0
ρ
)
eimϕϕ (z) , (6)
where m = 0;±1;±2; . . . is the magnetic quantum number, Jm
[
(χnρm)/ρ0ρ
]
is Bessel function of
the whole order, χnρm are roots of Bessel function (nρ is radial quantum number fixing the number
of Bessel function root at the fixed m).
70
Electron and hole spectra in quantum wire
Setting the wave function (6) in Schrodinger equation (4), the variables are separated and for
z-th term of wave function there is obtained the equation
∂2
∂z2
ϕ (z) + ϕ (z)
[
2µ (z)
~2
(E − U (ρ, ϕ, z)) −
χ2
nρm
ρ2
0
]
= 0. (7)
The solutions of equation (7) with notations
k2
0 =
2µ0
~2
E −
χ2
nρm
ρ2
0
, k2
1 =
2µ1
~2
(U0 − E) +
χ2
nρm
ρ2
0
for different parts of a nanosystem are as follows:
ϕ (z) =
A+ek0z, z < −h1 ,
B+eik1z + B−e−ik1z, −h1 6 z 6 0 ,
C+Ai[−ξ (z + zB)] + C−Bi[−ξ (z + zB)], 0 6 z 6 ∆ ,
D+eik1z + D−e−ik1z, ∆ 6 z 6 ∆ + h2 ,
E−e−k0z, z > ∆ + h2
(8)
here Ai (z) , Bi (z) are Airy functions of the first and second kind and
ξ =
(
2eFm
~2
)1/3
, zB =
E
eF
.
Using the conditions of a wave function and density of probability current continuity at the
media interfaces of a nanosystem (z = −h1, z = 0, z = ∆, z = ∆ + h2) and the condition of wave
function normalization
∞
∫
−∞
|ϕ (z)|
2
dz = 1, (9)
one can obtain analytical expressions for the coefficients A+, B±, C±,D±, E− (equation (8)) and
a dispersion equation for defining the electron energies in a nanoheterosystem (the respective
expressions are not presented because they are rather sophisticated). We should note that the
electron (hole) wave function and its energy are characterized by three quantum numbers: nρmnz
(Ψeh
nρmnz
(~r) , Eeh
nρmnz
). The axial quantum number (nz) numerates the solutions of dispersion equa-
tion at the fixed nρ and m quantum numbers.
3. Discussion of results
The computer calculations of electron (hole) energy spectrum is performed for the nanoheterosys-
tem created at the base of semiconductor crystals β − HgS (“1”) and β − CdS (“0”).
The results of calculating the electron and hole energy (Eeh
nρmnz
) (without the electric field) as
a function of the height of the second QD (h2) at the fixed nρ = 1, radius of QW:ρ0 = 8aHgS,
height of the other QD: h1 = 7aHgS and width of the potential barrier: ∆ = 2aCdS are presented
in figure 2.
Figure 2 proves that at h2 = 0 there are three (two) electron (hole) energy levels in a nanosys-
tem, coinciding (as it should be) with the levels arising in QD (HgS) with the height h1 embedded
into QW (CdS). The increase of h2 height causes the appearance of new energy levels becoming
smaller and creating anticrossings. They are caused by the splitting of the levels, having the origin
of both potential wells, accounting for their interaction through the potential barrier with finite
height and width.
71
O.Makhanets, A.Gryschyk, M.Dovganiuk
0 10 20 30 40 50
400
600
800
1000
1200
e
4
2
2
m
eV
104
113
112
111
103
102
E
103
E
111
E
101
m=0 n
z
=1n =1
h
2
, a
HgS
E
n
m
n
z ,
-600
-500
-400
-300
0,00
0 10 20 30 40 50
112
111
104
103
102
n
z
=1m=0n =1
2
E
111
2
E
101
h
m
eV
E
n
m
n
z ,
Figure 2. Dependences of electron and hole energies on the height of second QD (h2) at nρ = 1,
ρ0 = 8aHgS, h1 = 7aHgS, ∆ = 2aCdS, F = 0.
The number of energy levels is determined by the volume of quantum wells of the system. It
should be mentioned that at the equal geometric parameters, the number of electron energy states
is much bigger than of the hole ones. This is obviously caused by the fact that the depth of a
potential well for the electron (U e
0) is bigger than twice the respective magnitude for the hole (Uh
0 ).
From the general analysis of the dispersion equation it is clear that all electron and hole energy
levels (except the ground level) are twice degenerated in respect to the magnetic quantum number
m. Besides, as one can see in figure 2 there is a casual degeneration of energy levels because the
levels with different nz and m are crossing at the variation of h2.
The application of the electric field ~F does not qualitatively change the behavior of electron
(hole) energy spectrum but causes the shift of all energy levels into the region of smaller energies, to
the small shift of all anticrossings into the region of smaller magnitudes of h2 and to the varying of
the splitting of energy levels with equal symmetry ∆E
(e, h) n′
z
nρ m nz
. It is seen in figure 3 a,b, where there
are presented the dependences of electron (a) and hole (b) energies on the height of the quantum
dot h2 (in the range of 2 − 9 aHgS) at the fixed h1 = 7aHgS, ∆ = 2aCdS, ρ0 = 8aHgS when there is
no electric field F = 0 (solid curves) and for the electric field with the intensity F = 580 MV/m
(dash curves). In the figures one can see that the increase of intensity ~F causes the increase of the
splitting ∆E
(e, h) n′
z
nρ m nz
. The latter is bigger and increases faster with an increase of the anticrossings
in the energy scale of quantum well. It should be mentioned that the increase of the splitting for
the higher energy levels is capable of reaching 100 meV. Such a behavior of energy spectrum is
clear from physical viewpoint. Really, the increase of the electric field intensity causes the change of
potential barrier profile in such a way that, on the one hand, its effective thickness becomes smaller,
i.e., its transparency increases, obviously increasing the interaction between quantum wells and,
consequently, bringing to the increase of the splitting of the respective energy levels. On the other
hand, the increase of transparency effectively increases the quantum well volume (the toned region
72
Electron and hole spectra in quantum wire
2 3 4 5 6 7 8 9
400
600
800
1000
1200
a
e
2E'
101
2
E
101
E'
111
2
2E
111
4E'
103 4E
103
h
2
, a
HgS
m
eV
E
n
m
n
z ,
2 3 4 5 6 7 8 9
-600
-500
-400
-300
-200
b 2
E'
101
E
101
2
2E'
111 2E
111
h
2
, a
HgS
h
m
eV
E
n
m
n
z ,
Figure 3. Dependences of electron (a) and hole (b) energies on the height of quantum dot h2
at the fixed h1 = 7aHgS, ∆ = 2aCdS, ρ0 = 8aHgS when there is no electric field: F = 0 (solid
curves) and for the electric field with intensity F = 580 MV/m (dash curves).
0 6 z 6 ∆+h2 in figure 1). On the one hand, it decreases the electron and hole energy and, on the
other hand, causes the anticrossing arising at the smaller values of the second quantum dot (h2).
The final remark is that even an inessential shift of electron and hole energy levels into the
region of smaller energies at the increase of the electric field intensity ~F can essentially effect the
binding and exciting energy of exciton in the system under research. The detailed calculation and
analysis of exciton spectra are to be performed in future.
References
1. Aono T., Eto M., Kawamura K. J. Phys. Soc. Jpn., 1998, 67, 1860.
2. Georges A., Meir Y. Phys. Rev. Lett. 1999, 82, 3508.
3. Aguado R., Langreth D.C. Phys. Rev. Lett., 2000, 85, 1946.
4. Izumida W., Sakai O. Phys. Rev. B, 2000, 62, 10260.
5. Aono T., Eto M. Phys. Rev. B, 2001, 63, 125327.
6. Hu X., Sarma S.D. Phys. Rev. A, 2000, 61, 062301.
7. Burkard G., Seelig G., Loss D. Phys. Rev. B, 2000, 62, 2581.
8. Partoens B., Peeters F. Europhys. Lett., 2001, 56, 86.
9. Amaha S., Austing D.G., Tokura Y., Muraki K., Ono K., Tarucha S. Solid State Commun., 2001, 119,
183.
10. Bjork M.T., Thelander C., Hansen A.E., Jensen L.E., Larsson M.W., Wallenberg L.R., Samuelson L.
Nano Lett., 2004, 4, 1621.
11. Fasth C., Fuhrer A., Bjork M.T., Samuelson L. Nano Lett., 2005, 5, 1487.
12. Tkach N.V., Makhanets A.M. Fiz. Tv. Tela, 2005, 47, 550 (in Russian).
73
O.Makhanets, A.Gryschyk, M.Dovganiuk
Спектри електрона та дiрки у квантовому дротi з двома
квантовими точками в електричному полi
О.М.Маханець, А.М.Грищук, М.М.Довганюк
Чернiвецький нацiональний унiверситет iм. Юрiя Федьковича,
вул. Коцюбинського 2, 58012, Чернiвцi
Отримано 4 вересня 2006 р.
Дослiджено енергетичний спектр електрона i дiрки в складнiй наногетеросистемi, що складається з
двох цилiндричних напiвпровiдникових квантових точок, розташованих у напiвпровiдниковому кван-
товому дротi. Квантовi точки роздiленi шаром-бар’єром, до якого прикладене постiйне електричне
поле. Проаналiзовано залежностi енергiй електрона i дiрки вiд геометричних розмiрiв квантових
точок i величини напруженостi електричного поля.
Ключовi слова: квантовий дрiт, квантова точка, енергетичний спектр
PACS: 73.21.Fg, 73.21.Hb, 73.21.La
74
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